Suppose X is independent of Y, where X is any distribution, Y is Gaussian (0, $\sigma^2$). Do we have E(X|X+Y)=X+Y? Thank you!
2026-03-28 20:59:20.1774731560
Conditional Expection on Sum of 2 Random Variables E(X|X+Y)=X+Y?
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This is definetly not true in general. Note that $\mathbb{E}(X|X+Y)=X+Y$ implies that \begin{equation} \mathbb{E}(Y|X+Y)=0. \end{equation} For a counterexample, let $X$ and $Y$ be two independent Gaussian random variables. Then $\mathbb{E}(Y|X+Y)$ is again Gaussian (and not constantly equal to 0).